Let $G$ be a finite 2group of nilpotency class two such that $\frac{G}{Z(G)}=\{Z(G), aZ(G), bZ(G), abZ(G)\}\simeq C_{2}\times C_{2}$. Then do there exist a non inner automorphism $\alpha$ of $G$ such that $\alpha(a)\neq a$, $\alpha(b)\neq b$ and $\alpha(ab)\neq ab$ ? For example this true for $D_{8}$, dihedral group of order 8, or $Q_{8}$, generalized quaternion group of order 8.
